Paper - Multiple Regression SPA/ECO/WER 1

*The author of this computation has been verified*

R Software Module: rwasp_multipleregression.wasp (opens new window with default values)

Title produced by software: Multiple Regression

Date of computation: Sun, 07 Dec 2008 15:26:39 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2008/Dec/07/t1228689014nssn1c4df5yjr8r.htm/, Retrieved Sun, 07 Dec 2008 22:30:14 +0000

BibTeX entries for LaTeX users:

@Manual{KEY, author = {{YOUR NAME}}, publisher = {Office for Research Development and Education}, title = {Statistical Computations at FreeStatistics.org, URL http://www.freestatistics.org/blog/date/2008/Dec/07/t1228689014nssn1c4df5yjr8r.htm/}, year = {2008}, } @Manual{R, title = {R: A Language and Environment for Statistical Computing}, author = {{R Development Core Team}}, organization = {R Foundation for Statistical Computing}, address = {Vienna, Austria}, year = {2008}, note = {{ISBN} 3-900051-07-0}, url = {http://www.R-project.org}, }

Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)

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2008-11-27 13:41:43 [a2386b643d711541400692649981f2dc]  [reply]  test Post a new message

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No (this computation is public)

User-defined keywords:

Dataseries X:

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0 34 41 0 9 39 35 0 1 40 34 0 4 45 36 0 6 43 39 0 21 42 40 0 24 49 30 0 23 43 33 0 22 50 30 0 21 44 32 0 20 40 41 0 16 41 40 0 18 45 41 0 18 45 40 0 24 48 39 0 16 54 34 0 15 47 34 0 24 35 46 0 18 28 45 0 15 28 44 0 4 34 40 0 3 23 39 0 6 33 37 0 5 38 39 0 12 41 35 0 12 47 26 0 12 46 26 0 14 45 33 0 12 47 27 0 17 49 30 0 12 50 26 0 20 56 27 0 21 50 18 0 15 56 19 0 22 58 13 0 19 59 14 0 19 51 41 0 26 59 21 0 25 60 16 0 19 60 17 0 20 68 9 0 30 62 14 0 31 62 14 0 35 58 16 0 33 56 11 0 26 50 10 0 25 52 6 0 17 36 9 0 14 33 5 0 8 26 7 0 12 28 2 0 7 27 0 0 4 20 8 0 10 16 13 0 8 11 11 0 16 0 19 1 14 3 23 1 20 10 23 1 9 0 43 1 10 3 59 1

Output produced by software:

Enter (or paste) a matrix (table) containing all data (time) series. Every column represents a different variable and must be delimited by a space or Tab. Every row represents a period in time (or category) and must be delimited by hard returns. The easiest way to enter data is to copy and paste a block of spreadsheet cells. Please, do not use commas or spaces to seperate groups of digits! Summary of computational transaction Raw Input view raw input (R code) Raw Output view raw output of R engine Computing time 7 seconds R Server 'Herman Ole Andreas Wold' @ 193.190.124.10:1001 Multiple Linear Regression - Estimated Regression Equation Eco[t] = + 36.3120960039282 + 1.2533377170431Spa[t] -0.112704770277158Wer[t] -28.4495050600464Val[t] -1.80509600723871M1[t] -2.44516869039369M2[t] -1.53333249365160M3[t] + 4.07663116505126M4[t] + 3.79400612192671M5[t] -10.8669733372041M6[t] -9.56116647665295M7[t] -10.3559185676451M8[t] -5.14610644664224M9[t] -4.63456241309334M10[t] -3.26636837510925M11[t] -0.232995188815887t + e[t] Multiple Linear Regression - Ordinary Least Squares Variable Parameter S.D. T-STAT H0: parameter = 0 2-tail p-value 1-tail p-value (Intercept) 36.3120960039282 9.73421 3.7304 0.000544 0.000272 Spa 1.2533377170431 0.163881 7.6479 0 0 Wer -0.112704770277158 0.156012 -0.7224 0.473864 0.236932 Val -28.4495050600464 6.66714 -4.2671 0.000104 5.2e-05 M1 -1.80509600723871 5.664241 -0.3187 0.751475 0.375737 M2 -2.44516869039369 5.773942 -0.4235 0.674006 0.337003 M3 -1.53333249365160 5.830399 -0.263 0.793786 0.396893 M4 4.07663116505126 5.804933 0.7023 0.486209 0.243105 M5 3.79400612192671 5.811399 0.6529 0.517246 0.258623 M6 -10.8669733372041 5.729432 -1.8967 0.064445 0.032223 M7 -9.56116647665295 5.718851 -1.6719 0.101648 0.050824 M8 -10.3559185676451 5.831813 -1.7758 0.082688 0.041344 M9 -5.14610644664224 5.825686 -0.8833 0.38185 0.190925 M10 -4.63456241309334 5.759831 -0.8046 0.425357 0.212678 M11 -3.26636837510925 5.637508 -0.5794 0.565275 0.282637 t -0.232995188815887 0.132459 -1.759 0.085531 0.042765 Multiple Linear Regression - Regression Statistics Multiple R 0.887240501887896 R-squared 0.787195708190285 Adjusted R-squared 0.714648790527883 F-TEST (value) 10.850849816301 F-TEST (DF numerator) 15 F-TEST (DF denominator) 44 p-value 2.91507040728334e-10 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 8.81001791490602 Sum Squared Residuals 3415.12228908246 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 34 29.6531092265101 4.34689077348985 2 39 40.7363094295901 -1.73630942959008 3 40 31.5011534714486 8.49884652855136 4 45 40.4127255519106 4.5872744480894 5 43 42.0656664432249 0.934333556775106 6 42 45.8590527806476 -3.85905278064756 7 49 51.8189253062837 -2.81892530628368 8 43 49.1997259986011 -6.19972599860109 9 50 53.2613195245764 -3.26131952457643 10 44 52.061121111712 -8.06112111171202 11 40 50.9286393113427 -10.9286393113427 12 41 49.0613663997408 -8.06136639974081 13 45 49.4172458674953 -4.41724586749525 14 45 48.6568827658016 -3.65688276580155 15 48 56.9684548462635 -8.9684548462635 16 54 52.8822454311915 1.11775456880853 17 47 51.1132874822079 -4.11328748220793 18 35 46.1468950443233 -11.1468950443233 19 28 39.8123851840771 -11.8123851840771 20 28 35.1373295234169 -7.1373295234169 21 34 26.7782506492384 7.22174935076162 22 23 25.9161665472055 -2.91616654720546 23 33 31.0367880880573 1.96321191194273 24 38 32.5914140167532 5.40858598324678 25 41 39.7775059211090 1.22249407889105 26 47 39.9187809816325 7.0812190183675 27 46 40.5976219895587 5.40237801044129 28 45 47.6923325015918 -2.69233250159178 29 47 45.3462654572281 1.65373454277191 30 49 36.3808650836654 12.6191349163346 31 50 31.6378072512938 18.3621927487062 32 56 40.5240569375534 15.4759430624466 33 50 47.7685545192779 2.23144548072208 34 56 40.4143722914752 15.5856277085248 35 58 50.999163781608 7.00083621839198 36 59 50.1598190464949 8.84018095350507 37 51 45.0786990529571 5.92130094704295 38 59 55.233090605831 3.76690939416895 39 60 55.2221177480999 4.77788225190006 40 60 52.9663551454512 7.03364485454884 41 68 54.6057107927711 13.3942892072289 42 62 51.6815894638696 10.3184105361304 43 62 54.007738852648 7.99226114735204 44 58 57.767932900458 0.232067099541978 45 56 60.8015982499446 -4.80159824994457 46 50 52.419487845653 -2.41948784565304 47 52 52.7521680588868 -0.752168058886771 48 36 45.4207251980039 -9.42072519800386 49 33 40.0734399319286 -7.0734399319286 50 26 31.4549362171448 -5.45493621714482 51 28 37.7106519446292 -9.7106519446292 52 27 37.046341369855 -10.0463413698550 53 20 31.869069824568 -11.869069824568 54 16 23.9315976274941 -7.93159762749412 55 11 22.7231434056975 -11.7231434056975 56 0 2.37095463997056 -2.37095463997056 57 3 4.3902770569627 -1.39027705696270 58 10 12.1888522039543 -2.18885220395431 59 0 -2.71675923989475 2.71675923989475 60 3 -0.233324660992817 3.23332466099282 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 19 0.119021158761984 0.238042317523968 0.880978841238016 20 0.0730917517428404 0.146183503485681 0.92690824825716 21 0.0576911694502944 0.115382338900589 0.942308830549706 22 0.0358020014942963 0.0716040029885927 0.964197998505704 23 0.0173538196497068 0.0347076392994136 0.982646180350293 24 0.0116669325254087 0.0233338650508174 0.988333067474591 25 0.00869584210932499 0.0173916842186500 0.991304157890675 26 0.00364006867015412 0.00728013734030824 0.996359931329846 27 0.00196094650179824 0.00392189300359649 0.998039053498202 28 0.00924646371819135 0.0184929274363827 0.990753536281809 29 0.0478654222772944 0.0957308445545888 0.952134577722706 30 0.173813860243697 0.347627720487395 0.826186139756303 31 0.307793370582755 0.615586741165509 0.692206629417245 32 0.522709608942046 0.954580782115909 0.477290391057954 33 0.642240327800512 0.715519344398976 0.357759672199488 34 0.793339178869748 0.413321642260503 0.206660821130252 35 0.751954686676774 0.496090626646452 0.248045313323226 36 0.693666584917693 0.612666830164614 0.306333415082307 37 0.685194342539545 0.62961131492091 0.314805657460455 38 0.836183857698306 0.327632284603389 0.163816142301694 39 0.825778954567058 0.348442090865884 0.174221045432942 40 0.858806100017463 0.282387799965074 0.141193899982537 41 0.880858017770092 0.238283964459817 0.119141982229908 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 2 0.0869565217391304 NOK 5% type I error level 6 0.260869565217391 NOK 10% type I error level 8 0.347826086956522 NOK

Charts produced by software:

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Parameters (Session):

par1 = 2 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 2 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;

R code (references can be found in the software module):

library(lattice) library(lmtest) n25 <- 25 #minimum number of obs. for Goldfeld-Quandt test par1 <- as.numeric(par1) x <- t(y) k <- length(x[1,]) n <- length(x[,1]) x1 <- cbind(x[,par1], x[,1:k!=par1]) mycolnames <- c(colnames(x)[par1], colnames(x)[1:k!=par1]) colnames(x1) <- mycolnames #colnames(x)[par1] x <- x1 if (par3 == 'First Differences'){ x2 <- array(0, dim=c(n-1,k), dimnames=list(1:(n-1), paste('(1-B)',colnames(x),sep=''))) for (i in 1:n-1) { for (j in 1:k) { x2[i,j] <- x[i+1,j] - x[i,j] } } x <- x2 } if (par2 == 'Include Monthly Dummies'){ x2 <- array(0, dim=c(n,11), dimnames=list(1:n, paste('M', seq(1:11), sep =''))) for (i in 1:11){ x2[seq(i,n,12),i] <- 1 } x <- cbind(x, x2) } if (par2 == 'Include Quarterly Dummies'){ x2 <- array(0, dim=c(n,3), dimnames=list(1:n, paste('Q', seq(1:3), sep =''))) for (i in 1:3){ x2[seq(i,n,4),i] <- 1 } x <- cbind(x, x2) } k <- length(x[1,]) if (par3 == 'Linear Trend'){ x <- cbind(x, c(1:n)) colnames(x)[k+1] <- 't' } x k <- length(x[1,]) df <- as.data.frame(x) (mylm <- lm(df)) (mysum <- summary(mylm)) if (n > n25) { kp3 <- k + 3 nmkm3 <- n - k - 3 gqarr <- array(NA, dim=c(nmkm3-kp3+1,3)) numgqtests <- 0 numsignificant1 <- 0 numsignificant5 <- 0 numsignificant10 <- 0 for (mypoint in kp3:nmkm3) { j <- 0 numgqtests <- numgqtests + 1 for (myalt in c('greater', 'two.sided', 'less')) { j <- j + 1 gqarr[mypoint-kp3+1,j] <- gqtest(mylm, point=mypoint, alternative=myalt)$p.value } if (gqarr[mypoint-kp3+1,2] < 0.01) numsignificant1 <- numsignificant1 + 1 if (gqarr[mypoint-kp3+1,2] < 0.05) numsignificant5 <- numsignificant5 + 1 if (gqarr[mypoint-kp3+1,2] < 0.10) numsignificant10 <- numsignificant10 + 1 } gqarr } bitmap(file='test0.png') plot(x[,1], type='l', main='Actuals and Interpolation', ylab='value of Actuals and Interpolation (dots)', xlab='time or index') points(x[,1]-mysum$resid) grid() dev.off() bitmap(file='test1.png') plot(mysum$resid, type='b', pch=19, main='Residuals', ylab='value of Residuals', xlab='time or index') grid() dev.off() bitmap(file='test2.png') hist(mysum$resid, main='Residual Histogram', xlab='values of Residuals') grid() dev.off() bitmap(file='test3.png') densityplot(~mysum$resid,col='black',main='Residual Density Plot', xlab='values of Residuals') dev.off() bitmap(file='test4.png') qqnorm(mysum$resid, main='Residual Normal Q-Q Plot') qqline(mysum$resid) grid() dev.off() (myerror <- as.ts(mysum$resid)) bitmap(file='test5.png') dum <- cbind(lag(myerror,k=1),myerror) dum dum1 <- dum[2:length(myerror),] dum1 z <- as.data.frame(dum1) z plot(z,main=paste('Residual Lag plot, lowess, and regression line'), ylab='values of Residuals', xlab='lagged values of Residuals') lines(lowess(z)) abline(lm(z)) grid() dev.off() bitmap(file='test6.png') acf(mysum$resid, lag.max=length(mysum$resid)/2, main='Residual Autocorrelation Function') grid() dev.off() bitmap(file='test7.png') pacf(mysum$resid, lag.max=length(mysum$resid)/2, main='Residual Partial Autocorrelation Function') grid() dev.off() bitmap(file='test8.png') opar <- par(mfrow = c(2,2), oma = c(0, 0, 1.1, 0)) plot(mylm, las = 1, sub='Residual Diagnostics') par(opar) dev.off() if (n > n25) { bitmap(file='test9.png') plot(kp3:nmkm3,gqarr[,2], main='Goldfeld-Quandt test',ylab='2-sided p-value',xlab='breakpoint') grid() dev.off() } load(file='createtable') a<-table.start() a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Estimated Regression Equation', 1, TRUE) a<-table.row.end(a) myeq <- colnames(x)[1] myeq <- paste(myeq, '[t] = ', sep='') for (i in 1:k){ if (mysum$coefficients[i,1] > 0) myeq <- paste(myeq, '+', '') myeq <- paste(myeq, mysum$coefficients[i,1], sep=' ') if (rownames(mysum$coefficients)[i] != '(Intercept)') { myeq <- paste(myeq, rownames(mysum$coefficients)[i], sep='') if (rownames(mysum$coefficients)[i] != 't') myeq <- paste(myeq, '[t]', sep='') } } myeq <- paste(myeq, ' + e[t]') a<-table.row.start(a) a<-table.element(a, myeq) a<-table.row.end(a) a<-table.end(a) table.save(a,file='mytable1.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a,hyperlink('http://www.xycoon.com/ols1.htm','Multiple Linear Regression - Ordinary Least Squares',''), 6, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'Variable',header=TRUE) a<-table.element(a,'Parameter',header=TRUE) a<-table.element(a,'S.D.',header=TRUE) a<-table.element(a,'T-STAT<br />H0: parameter = 0',header=TRUE) a<-table.element(a,'2-tail p-value',header=TRUE) a<-table.element(a,'1-tail p-value',header=TRUE) a<-table.row.end(a) for (i in 1:k){ a<-table.row.start(a) a<-table.element(a,rownames(mysum$coefficients)[i],header=TRUE) a<-table.element(a,mysum$coefficients[i,1]) a<-table.element(a, round(mysum$coefficients[i,2],6)) a<-table.element(a, round(mysum$coefficients[i,3],4)) a<-table.element(a, round(mysum$coefficients[i,4],6)) a<-table.element(a, round(mysum$coefficients[i,4]/2,6)) a<-table.row.end(a) } a<-table.end(a) table.save(a,file='mytable2.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Regression Statistics', 2, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Multiple R',1,TRUE) a<-table.element(a, sqrt(mysum$r.squared)) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'R-squared',1,TRUE) a<-table.element(a, mysum$r.squared) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Adjusted R-squared',1,TRUE) a<-table.element(a, mysum$adj.r.squared) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'F-TEST (value)',1,TRUE) a<-table.element(a, mysum$fstatistic[1]) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'F-TEST (DF numerator)',1,TRUE) a<-table.element(a, mysum$fstatistic[2]) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'F-TEST (DF denominator)',1,TRUE) a<-table.element(a, mysum$fstatistic[3]) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'p-value',1,TRUE) a<-table.element(a, 1-pf(mysum$fstatistic[1],mysum$fstatistic[2],mysum$fstatistic[3])) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Residual Statistics', 2, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Residual Standard Deviation',1,TRUE) a<-table.element(a, mysum$sigma) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Sum Squared Residuals',1,TRUE) a<-table.element(a, sum(myerror*myerror)) a<-table.row.end(a) a<-table.end(a) table.save(a,file='mytable3.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Actuals, Interpolation, and Residuals', 4, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Time or Index', 1, TRUE) a<-table.element(a, 'Actuals', 1, TRUE) a<-table.element(a, 'Interpolation<br />Forecast', 1, TRUE) a<-table.element(a, 'Residuals<br />Prediction Error', 1, TRUE) a<-table.row.end(a) for (i in 1:n) { a<-table.row.start(a) a<-table.element(a,i, 1, TRUE) a<-table.element(a,x[i]) a<-table.element(a,x[i]-mysum$resid[i]) a<-table.element(a,mysum$resid[i]) a<-table.row.end(a) } a<-table.end(a) table.save(a,file='mytable4.tab') if (n > n25) { a<-table.start() a<-table.row.start(a) a<-table.element(a,'Goldfeld-Quandt test for Heteroskedasticity',4,TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'p-values',header=TRUE) a<-table.element(a,'Alternative Hypothesis',3,header=TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'breakpoint index',header=TRUE) a<-table.element(a,'greater',header=TRUE) a<-table.element(a,'2-sided',header=TRUE) a<-table.element(a,'less',header=TRUE) a<-table.row.end(a) for (mypoint in kp3:nmkm3) { a<-table.row.start(a) a<-table.element(a,mypoint,header=TRUE) a<-table.element(a,gqarr[mypoint-kp3+1,1]) a<-table.element(a,gqarr[mypoint-kp3+1,2]) a<-table.element(a,gqarr[mypoint-kp3+1,3]) a<-table.row.end(a) } a<-table.end(a) table.save(a,file='mytable5.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a,'Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity',4,TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'Description',header=TRUE) a<-table.element(a,'# significant tests',header=TRUE) a<-table.element(a,'% significant tests',header=TRUE) a<-table.element(a,'OK/NOK',header=TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'1% type I error level',header=TRUE) a<-table.element(a,numsignificant1) a<-table.element(a,numsignificant1/numgqtests) if (numsignificant1/numgqtests < 0.01) dum <- 'OK' else dum <- 'NOK' a<-table.element(a,dum) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'5% type I error level',header=TRUE) a<-table.element(a,numsignificant5) a<-table.element(a,numsignificant5/numgqtests) if (numsignificant5/numgqtests < 0.05) dum <- 'OK' else dum <- 'NOK' a<-table.element(a,dum) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'10% type I error level',header=TRUE) a<-table.element(a,numsignificant10) a<-table.element(a,numsignificant10/numgqtests) if (numsignificant10/numgqtests < 0.1) dum <- 'OK' else dum <- 'NOK' a<-table.element(a,dum) a<-table.row.end(a) a<-table.end(a) table.save(a,file='mytable6.tab') }

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